This page intentionally left blank CAMBRIDGE STUDIES IN ADVANCED MATHEMATICS 105 EDITORIAL BOARD B. BOLLOBAS, W. FULTON, A. KATOK, F. KIRWAN, P. SARNAK, B. SIMON, B. TOTARO ADDITIVE COMBINATORICS Additive combinatorics is the theory of counting additive structures in sets. This theory has seen exciting developments and dramatic changes in direction in recent years, thanks to its connections with areas such as number theory, ergodic theory and graph theory. This graduate level textbook will allow students and researchers easy entry into this fascinating field. Here, for the first time, the authors bring together, in a self-contained and systematic manner, the many different tools and ideas that are used in the modern theory, presenting them in an accessible, coherent, and intuitively clear manner, and providing immediate applications to problems in additive combinatorics. The power of these tools is well demonstrated in the presentation of recent advances such as the Green-Tao theorem on arithmetic progressions and Erd˝os distance problems, and the developing field of sum-product estimates. The text is supplemented by a large number of exercises and new material. Terence Tao is a professor in the Department of Mathematics at the University of California, Los Angeles. Van Vu is a professor in the Department of Mathematics at Rutgers University, New Jersey. CAMBRIDGE STUDIES IN ADVANCED MATHEMATICS Editorial Board: B. Bollob´as, W. Fulton, A. Katok, F. Kirwan, P. Sarnak, B. Simon, B. Totaro Au the title, listed below can be obtained from good booksellers or from Cambridge University Press for a complete listing visit www.cambridge.org/uk/series/&Series.asp?code=CSAM. 49 R. Stanley Enumerative combinatorics I 50 I. Porteous Clifford algebras and the classical groups 51 M. Audin Spinning tops 52 V. Jurdjevic Geometric control theory 53 H. Volklein Groups as Galois groups 54 J. Le Potier Lectures on vector bundles 55 D. Bump Automorphic forms and representations 56 G. Laumon Cohomology of Drinfeld modular varieties II 57 D.M. Clark & B.A. Davey Natural dualities for the working algebraist 58 J. McCleary A user’s guide to spectral sequences II 59 P. Taylor Practical foundations of mathematics 60 M.P. Brodmann & R.Y. Sharp Local cohomology 61 J.D. Dixon et al. Analytic pro-P groups 62 R. Stanley Enumerative combinatorics II 63 R.M. Dudley Uniform central limit theorems 64 J. Jost & X. Li-Jost Calculus of variations 65 A.J. Berrick & M.E. Keating An introduction to rings and modules 66 S. Morosawa Holomorphic dynamics 67 A.J. Berrick & M.E. Keating Categories and modules with K-theory in view 68 K. Sato Levy processes and infinitely divisible distributions 69 H. Hida Modular forms and Galois cohomology 70 R. Iorio & V. Iorio Fourier analysis and partial differential equations 71 R. Blei Analysis in integer and fractional dimensions 72 F. Borceaux & G. Janelidze Galois theories 73 B. Bollobas Random graphs 74 R.M. Dudley Real analysis and probability 75 T. Sheil-Small Complex polynomials 76 C. Voisin Hodge theory and complex algebraic geometry I 77 C. Voisin Hodge theory and complex algebraic geometry II 78 V. Paulsen Completely bounded maps and operator algebras 79 F. Gesztesy & H. Holden Soliton Equations and their Algebro-Geometric Solutions Volume 1 81 Shigeru Mukai An Introduction to Invariants and Moduli 82 G. Tourlakis Lectures in logic and set theory I 83 G. Tourlakis Lectures in logic and set theory II 84 R.A. Bailey Association Schemes 85 James Carlson, Stefan M¨uller-Stach, & Chris Peters Period Mappings and Period Domains 86 J.J. Duistermaat & J.A.C. Kolk Multidimensional Real Analysis I 87 J.J. Duistermaat & J.A.C. Kolk Multidimensional Real Analysis II 89 M. Golumbic & A.N. Trenk Tolerance Graphs 90 L.H. Harper Global Methods for Combinatorial Isoperimetric Problems 91 I. Moerdijk & J. Mrcun Introduction to Foliations and Lie Groupoids 92 J´anos Koll´ar, Karen E. Smith, & Alessio Corti Rational and Nearly Rational Varieties 93 David Applebaum L ´ evy Processes and Stochastic Calculus 95 Martin Schechter An Introduction to Nonlinear Analysis Additive Combinatorics TERENCE TAO, VAN VU cambridge university press Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo Cambridge University Press The Edinburgh Building, Cambridge cb2 2ru, UK First published in print format isbn-13 978-0-521-85386-6 isbn-13 978-0-511-24530-5 © Cambridge University Press 2006 2006 Information on this title: www.cambrid g e.or g /9780521853866 This publication is in copyright. Subject to statutory exception and to the provision of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. isbn-10 0-511-24530-0 isbn-10 0-521-85386-9 Cambridge University Press has no responsibility for the persistence or accuracy of urls for external or third-party internet websites referred to in this publication, and does not guarantee that any content on such websites is, or will remain, accurate or appropriate. Published in the United States of America by Cambridge University Press, New York www.cambridge.org hardback eBook (EBL) eBook (EBL) hardback To our families Contents Prologue page xi 1 The probabilistic method 1 1.1 The first moment method 2 1.2 The second moment method 6 1.3 The exponential moment method 9 1.4 Correlation inequalities 19 1.5 The Lov´asz local lemma 23 1.6 Janson’s inequality 27 1.7 Concentration of polynomials 33 1.8 Thin bases of higher order 37 1.9 Thin Waring bases 42 1.10 Appendix: the distribution of the primes 45 2 Sum set estimates 51 2.1 Sum sets 54 2.2 Doubling constants 57 2.3 Ruzsa distance and additive energy 59 2.4 Covering lemmas 69 2.5 The Balog–Szemer´edi–Gowers theorem 78 2.6 Symmetry sets and imbalanced partial sum sets 83 2.7 Non-commutative analogs 92 2.8 Elementary sum-product estimates 99 3 Additive geometry 112 3.1 Additive groups 113 3.2 Progressions 119 3.3 Convex bodies 122 vii viii Contents 3.4 The Brunn–Minkowski inequality 127 3.5 Intersecting a convex set with a lattice 130 3.6 Progressions and proper progressions 143 4 Fourier-analytic methods 149 4.1 Basic theory 150 4.2 L p theory 156 4.3 Linear bias 160 4.4 Bohr sets 165 4.5 (p) constants, B h [g] sets, and dissociated sets 172 4.6 The spectrum of an additive set 181 4.7 Progressions in sum sets 189 5 Inverse sum set theorems 198 5.1 Minimal size of sum sets and the e-transform 198 5.2 Sum sets in vector spaces 211 5.3 Freiman homomorphisms 220 5.4 Torsion and torsion-free inverse theorems 227 5.5 Universal ambient groups 233 5.6 Freiman’s theorem in an arbitrary group 239 6 Graph-theoretic methods 246 6.1 Basic Notions 247 6.2 Independent sets, sum-free subsets, and Sidon sets 248 6.3 Ramsey theory 254 6.4 Proof of the Balog–Szemer´edi–Gowers theorem 261 6.5 Pl¨unnecke’s theorem 267 7 The Littlewood–Offord problem 276 7.1 The combinatorial approach 277 7.2 The Fourier-analytic approach 281 7.3 The Ess´een concentration inequality 290 7.4 Inverse Littlewood–Offord results 292 7.5 Random Bernoulli matrices 297 7.6 The quadratic Littlewood–Offord problem 304 8 Incidence geometry 308 8.1 The crossing number of a graph 308 8.2 The Szemer´edi–Trotter theorem 311 8.3 The sum-product problem in R 315 8.4 Cell decompositions and the distinct distances problem 319 8.5 The sum-product problem in other fields 325 [...]... theorem and the Pl¨ nnecke inequale u ities Two other important tools from graph theory, namely the crossing number inequality and the Szemer´ di regularity lemma, will also be covered in Chapter e 8 and Sections 10.6, 11.6 respectively In Chapter 7 we view sum sets from the perspective of random walks, and give some classical and recent results concerning the distribution of these sum sets, and in... than or equal to x Landau asymptotic notation Let n be a positive variable (usually taking values on N, Z+ , R≥0 , or R+ , and often assumed to be large) and let f (n) and g(n) be real-valued functions of n r g(n) = O( f (n)) means that f is non-negative, and there is a positive constant C such that |g(n)| ≤ C f (n) for all n r g(n) = ( f (n)) means that f, g are non-negative, and there is a positive... I(E) = 1 if E occurs and 0 otherwise) If E, F are events, we use E ∧ F to denote the event that E, F both ¯ hold, E ∨ F to denote the event that at least one of E, F hold, and E to denote the event that E does not hold In this chapter all random variables will be assumed to be real-valued (and usually denoted by X or Y ) or set-valued (and usually denoted by B) If X is a real-valued random variable with... group Z p rather than the integers Z, and observe that a subset B of A will be sum-free in Z p if and only if 1 it is sum-free in Z Now choose a random number x ∈ Z p \{0} uniformly, and form the random set B := A ∩ (x · [k + 1, 2k + 1]) = {a ∈ A : x −1 a ∈ {k + 1, , 2k + 1}} Since [k + 1, 2k + 1] is sum-free in Z p , we see that x · [k + 1, 2k + 1] is too, and thus B is a sum-free subset of A We... “shared additive structure” (e.g A and B are progressions with the same step size v); we invite the reader to devise analogs of the above criteria to capture this concept Making the above heuristics precise and rigorous will require some work, and in fact will occupy large parts of Chapters 2, 3, 4, 5, 6 In deriving these basic tools of the field, we shall need to develop and combine techniques from elementary... non-negative and both g(n) = O( f (n)) and g(n) = ( f (n)) hold; that is, there are positive constants c and C such that c f (n) ≥ g(n) ≥ C f (n) for all n r g(n) = o n→∞ ( f (n)) means that f is non-negative and g(n) = O(a(n) f (n)) for some a(n) which tends to zero as n → ∞; if f is strictly positive, this is equivalent to limn→∞ g(n)/ f (n) = 0 r g(n) = ω n→∞ ( f (n)) means that f, g are non-negative and. .. them in a self-contained and introductory manner, and illustrate their application to problems in additive combinatorics Many aspects of this material have already been covered in other papers and texts (and in particular several earlier books [168], [257], [116] have focused on some of the aspects of additive combinatorics), but this book attempts to present as many perspectives and techniques as possible... Nathanson, Imre Ruzsa, Roman Sasyk, and Benny Sudakov for helpful comments and corrections, Prologue xv and to the Australian National University and the University of Edinburgh for their hospitality while portions of this book were being written Parts of this work were inspired by the lecture notes of Ben Green [144], the expository article of Imre Ruzsa [297], and the book by Melvyn Nathanson [257]... 1.2.4 1.2.5 1.2.6 When does equality hold in Chebyshev’s inequality? If X and Y are two random variables, verify the Cauchy–Schwarz inequality |Cov(X, Y )| ≤ Var(X )1/2 Var(Y )1/2 and the triangle inequality Var(X + Y )1/2 ≤ Var(X )1/2 + Var(Y )1/2 When does equality occur? Prove (1.10) If φ : R → R is a convex function and X is a random variable, verify Jensen’s inequality E(φ(X )) ≤ φ(E(X )) If φ is... E(X )/σ As an immediate consequence of Corollary 1.9 and (1.4) we obtain the following concentration of measure property for the distribution of certain types of random sets Corollary 1.10 Let A be a set (possibly infinite), and let B ⊂ A be a random subset of A with the property that the events a ∈ B are independent for every a ∈ A Then for any > 0 and any finite A ⊆ A we have P ||B ∩ A | − pa | ≥ a∈A . from the perspective of random walks, and give some classical and recent results concerning the distribution of these sum sets, and in particular recent applications to random matrices. Last, but. introduction to rings and modules 66 S. Morosawa Holomorphic dynamics 67 A.J. Berrick & M.E. Keating Categories and modules with K-theory in view 68 K. Sato Levy processes and infinitely divisible. Bollobas Random graphs 74 R.M. Dudley Real analysis and probability 75 T. Sheil-Small Complex polynomials 76 C. Voisin Hodge theory and complex algebraic geometry I 77 C. Voisin Hodge theory and complex